details

property	value
status	complete
benchmark	10.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n171.star.cs.uiowa.edu
space	Various_04

run statistics

property	value
solver	AProVE23
configuration	certified
runtime (wallclock)	2.19877386093 seconds
cpu usage	5.872895625
max memory	3.50040064E8

stage attributes

key	value
certification-result	CERTIFIED
starexec-result	YES
certification-time	0.02

output

/export/starexec/sandbox/solver/bin/starexec_run_certified /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES <?xml version="1.0" encoding="UTF-8" standalone="no"?><?xml-stylesheet type="text/xsl" href="cpfHTML.xsl"?><certificationProblem xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" 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></component><component><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>sum</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>:</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>+</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>++</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>sum</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>++</name><arg><funapp><name>nil</name></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><var>ys</var></rhs></rule><rule><lhs><funapp><name>++</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>++</name><arg><funapp><name>nil</name></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><var>ys</var></rhs></rule><rule><lhs><funapp><name>++</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>sum</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>:</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>+</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>++</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>sum</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>++</name><arg><funapp><name>nil</name></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><var>ys</var></rhs></rule><rule><lhs><funapp><name>++</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>++</name><arg><funapp><name>nil</name></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><var>ys</var></rhs></rule><rule><lhs><funapp><name>++</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><monoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>sum</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>:</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>+</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>++</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>sum</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>++</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule></rules></trs><dpProof><monoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>sum</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>:</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>+</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>++</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><sharp><name>sum</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules><rule><lhs><funapp><name>+</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule></rules></trs><dpProof><switchInnermostProc><wcrProof><joinableCriticalPairsBFS>20</joinableCriticalPairsBFS></wcrProof><dpProof><usableRulesProc><usableRules><rules/></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></innermostLhss><dpProof><redPairProc><orderingConstraintProof><redPair><knuthBendixOrder><w0>1</w0><precedenceWeight><precedenceWeightEntry><name>:</name><arity>2</arity><precedence>1</precedence><weight>2</weight></precedenceWeightEntry><precedenceWeightEntry><name>sum</name><arity>1</arity><precedence>0</precedence><weight>1</weight></precedenceWeightEntry></precedenceWeight><argumentFilter><argumentFilterEntry><sharp><name>sum</name></sharp><arity>1</arity><collapsing>1</collapsing></argumentFilterEntry><argumentFilterEntry><name>++</name><arity>2</arity><collapsing>2</collapsing></argumentFilterEntry><argumentFilterEntry><name>:</name><arity>2</arity><nonCollapsing/></argumentFilterEntry><argumentFilterEntry><name>sum</name><arity>1</arity><nonCollapsing/></argumentFilterEntry></argumentFilter></knuthBendixOrder></redPair></orderingConstraintProof><dps><rules/></dps><dpProof><pIsEmpty/></dpProof></redPairProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></switchInnermostProc></dpProof></monoRedPairProc></dpProof></monoRedPairProc></dpProof></monoRedPairUrProc></dpProof></monoRedPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>avg</name></sharp><arg><var>xs</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><funapp><name>hd</name><arg><funapp><name>sum</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><funapp><name>length</name><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>:</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>+</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>:</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>sum</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>:</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules><rule><lhs><funapp><name>+</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><name>+</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><switchInnermostProc><wcrProof><joinableCriticalPairsBFS>20</joinableCriticalPairsBFS></wcrProof><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>+</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>sum</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>:</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules/></dps><trs><rules><rule><lhs><funapp><name>+</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><name>+</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>+</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>+</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><pIsEmpty/></dpProof></monoRedPairUrProc></dpProof></switchInnermostProc></dpProof></monoRedPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>quot</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><funapp><name>-</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><redPairUrProc><orderingConstraintProof><redPair><knuthBendixOrder><w0>1</w0><precedenceWeight><precedenceWeightEntry><name>s</name><arity>1</arity><precedence>0</precedence><weight>1</weight></precedenceWeightEntry><precedenceWeightEntry><name>0</name><arity>0</arity><precedence>1</precedence><weight>1</weight></precedenceWeightEntry></precedenceWeight><argumentFilter><argumentFilterEntry><sharp><name>quot</name></sharp><arity>2</arity><collapsing>1</collapsing></argumentFilterEntry><argumentFilterEntry><name>s</name><arity>1</arity><nonCollapsing><position>1</position></nonCollapsing></argumentFilterEntry><argumentFilterEntry><name>-</name><arity>2</arity><collapsing>1</collapsing></argumentFilterEntry><argumentFilterEntry><name>0</name><arity>0</arity><nonCollapsing/></argumentFilterEntry></argumentFilter></knuthBendixOrder></redPair></orderingConstraintProof><dps><rules/></dps><usableRules><rules><rule><lhs><funapp><name>-</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>-</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>-</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>-</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></usableRules><dpProof><pIsEmpty/></dpProof></redPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>+</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><funapp><name>++</name><arg><var>xs</var></arg><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>++</name></sharp><arg><var>xs</var></arg><arg><funapp><name>sum</name><arg><funapp><name>:</name><arg><var>x</var></arg><arg><funapp><name>:</name><arg><var>y</var></arg><arg><var>ys</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>quot</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>-</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>avg</name></sharp><arg><var>xs</var></arg></funapp></lhs><rhs><funapp><sharp><name>length</name></sharp><arg><var>xs</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>+</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>+</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>+</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>+</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>+</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules/></usableRules><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>+</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>+</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge><edge><position>2</position><strict>false</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></monoRedPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>++</name></sharp><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><sharp><name>++</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>:</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>++</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>++</name></sharp><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><sharp><name>++</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules/></usableRules><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>++</name></sharp><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg><arg><var>ys</var></arg></funapp></lhs><rhs><funapp><sharp><name>++</name></sharp><arg><var>xs</var></arg><arg><var>ys</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge><edge><position>2</position><strict>false</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></monoRedPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>-</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>-</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>-</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>-</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>-</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules/></usableRules><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>-</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>-</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge><edge><position>2</position><strict>true</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></monoRedPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>length</name></sharp><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>length</name></sharp><arg><var>xs</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>:</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>length</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>length</name></sharp><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>length</name></sharp><arg><var>xs</var></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules/></usableRules><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>length</name></sharp><arg><funapp><name>:</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>length</name></sharp><arg><var>xs</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></monoRedPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>avg</name></sharp><arg><var>xs</var></arg></funapp></lhs><rhs><funapp><sharp><name>hd</name></sharp><arg><funapp><name>sum</name><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></dpTrans></trsTerminationProof></proof><origin><proofOrigin><tool><name>AProVE</name><version>AProVE Commit ID: 363435ff5b19173ec8a83b45cfdec073678dabcc jan-christoph 20230815 unpublished </version><strategy>Statistics for single proof: 100.00 % (25 real / 0 unknown / 0 assumptions / 25 total proof steps)</strategy><url>http://aprove.informatik.rwth-aachen.de</url></tool><toolUser><firstName>John</firstName><lastName>Doe</lastName></toolUser></proofOrigin><inputOrigin/></origin></certificationProblem> popout

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